Adrian Bejan(Editor), Allan D. Kraus (Editor). Heat transfer Handbok (776115), страница 70
Текст из файла (страница 70)
(6.51). From thisθAW (0) ≡ rc = b(Pr) Pr1/2(for gases)where rc is called the recovery factor. Using this, the adiabatic wall temperature iscalculated asTAW = T∞ + rcU22cpFor low velocities this can be approximated as[462], (24)TAW = T∞The temperature profile for various choices of T0 is shown in Fig. 6.9 from Gebhart(1971).The heat flux can be written asq0 ∂T U 2 U 1/2 = −k= −k[θAW (0) + C1 φ (0)]∂y y=02cp νx 1/2U1/3= k(T0 − TAW ) 0.332· Prνx———8.63919pt PgVar———Long PagePgEnds: TEX(6.54)Figure 6.9 Boundary layer temperature profiles with viscous dissipation. (From Gebhart,1971.)BOOKCOMP, Inc.
— John Wiley & Sons / Page 462 / 2nd Proofs / Heat Transfer Handbook / BejanLines: 1085 to 1137[462], (24)HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW123456789101112131415161718192021222324252627282930313233343536373839404142434445463and with the definitionq0 = hx (T0 − TAW )the Nusselt number becomes the well-known Pohlhausen (1921) solution:1/3Nu = 0.332Re1/2x · Prwhere the properties can be evaluated at the reference temperature recommended byEckert:T ∗ = T∞ + (TW − T∞ ) + 0.22(TAW − T∞ )[463], (25)6.4.7 Integral Solutions for a Flat Plate Boundary Layerwith Unheated Starting LengthConsider the configuration illustrated in Fig. 6.10. The solution for this configurationcan be used as a building block for an arbitrarily varying surface temperature wherea similarity solution does not exist. Assuming steady flow at constant properties andno viscous dissipation, the boundary layer momentum and energy equations can beintegrated across the respective boundary layers to yieldddxddx(6.55)∂T u(T∞ − T ) dy = α∂y y=0(6.56)u(U − u) dy = ν0 δT0∂u ∂y y=0 δ␦x0␦TxT0Figure 6.10 Hydrodynamic and thermal boundary layer development along a flat plate withan unheated starting length in a uniform stream.BOOKCOMP, Inc.
— John Wiley & Sons / Page 463 / 2nd Proofs / Heat Transfer Handbook / Bejan———0.47308pt PgVar———Long Page* PgEnds: Eject[463], (25)To integrate eqs. (6.55) and (6.56), approximate profiles for tangential velocity andtemperature across the boundary layer must be defined. For example, a cubic parabolaprofile of the type T = a + by + cy 2 + dy 3 can be employed, with the conditionsU, T ⬁Lines: 1137 to 1175464123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWSy(T = T0 ) = 0(6.57a)y(T = T∞ ) = δT∂T =0∂y T =T∞(6.57b)(6.57c)By evaluating the energy equation at the surface, an additional condition can bedeveloped:∂ 2 T =0∂y 2 y=0Using these boundary conditions, the temperature profile can be determined asT − T03 y1=−T∞ − T 02 δT2yδT3[464], (26)(6.58)Lines: 1175 to 1233Polynomials of higher order can be selected, with the additional conditions determined by prescribing additional higher-order derivatives set to zero at the edge of theboundary layer.
In a similar manner, a cubic velocity profile can be determined byprescribing———1.94324pt PgVar———Long Page* PgEnds: Ejectu(y = 0) = v(y = 0) = 0[464], (26)u(y = δ) = u∞∂u =0∂y y=δand determining∂ 2 u =0∂y 2 y=0This results inu3y1 y 3=−U2δ2 δ(6.59)These profiles are next substituted into the integral energy equation and the integration carried out. Assuming that Pr > 1, the upper limit needs only to be extendedto y = δT , because beyond this, θ = T0 − T = T − T∞ = θ∞ and the integrand iszero. In addition, defining, r = δT /δ, it noted that δT3 23 4(6.60)r −r(θ∞ − θ)u dy = θ∞ U δ202800BOOKCOMP, Inc. — John Wiley & Sons / Page 464 / 2nd Proofs / Heat Transfer Handbook / BejanHEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW123456789101112131415161718192021222324252627282930313233343536373839404142434445465Because r < 1, the second term may be neglected in comparison to the first.
Now ris a function of x and the integral of eq. (6.60) may be put into the integral energyequation of eq. (6.56), yielding2r 2 δ210αdrdδ+ r 3δ=dxdxU(6.61)From the integral momentum equation, δ(x) can be determined asδ(x) = 4.64 νx 1/2UThis yieldsr 3 + 4r 2 xdr13=dx14Pr[465], (27)which can be solved to yieldLines: 1233 to 1299 x 3/4 1/310r=1−x1.026Pr1/3———-0.20381pt PgVar———where x0 is the unheated starting length. The heat transfer coefficient and the NusseltLong Pagenumber are then* PgEnds: Eject−k(∂T /∂y)|y=0k ∂θ 3 k3 kh====T0 − T ∞θ∞ ∂y y=02 δT2 rδ[465], (27)orPr1/3h = 0.332k[1 − (x0 /x)3/4 ]1/3Uνx1/2(6.62)andNu =0.332Pr1/2 · Re1/2x[1 − (x0 /x)3/4 ]1/3(6.63)Arbitrarily Varying Surface Temperature The foregoing results can easily begeneralized for any surface temperature variation of the typeT0 = T∞ + A +∞Bn x nn=1For a single step,T0 − T= θ(x0 , x, y)T0 − T ∞BOOKCOMP, Inc.
— John Wiley & Sons / Page 465 / 2nd Proofs / Heat Transfer Handbook / Bejan(6.64)466123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWSand for an arbitrary variation,xT0 − T∞ =k[1 − θ(x0 , x, y)]0q0 = k0x∂Tdx0 +[1 − θ(x0i , x, y)] ∆T0,i∂x0i=1(6.65)k ∂θ(x0 , x, 0)∂θ(x0 , x, 0) dT0idx0 +∆T0,i∂ydx0∂yi=1(6.66)This procedure using the integral momentum equation can also be generalized toa turbulent boundary layer with an arbitrary surface temperature variation.6.4.8 Two-Dimensional Nonsimilar Flows[466], (28)When similarity conditions do not apply, as in the case of an unheated starting lengthplate, two classes of approaches exist for solution of the governing equations.
Theintegral method results in an ordinary differential equation with the downstreamLines: 1299 to 1339coordinate x as the independent variable and parameters associated with the body———profile shape and the various boundary layer thicknesses as the dependent variable.-0.54802ptPgVarGenerally, such solutions result in correlation relationships that have a limited range———of applicability. These are typically much faster to compute.Normal PageDifferential methods solve the partial differential equations describing numericallythe conservation of mass, force momentum balance, and conservation of energy. * PgEnds: EjectThese equations are discretized over a number of control volumes, resulting in a setof algebraic equations that are solved simultaneously using numerical techniques.[466], (28)These solutions provide the detailed velocity, pressure, temperature, and density fieldsfor compressible flows.
The heat transfer rates from various surfaces can also bedetermined from these results. In the 1980s and 1990s, the computational hardwarecapabilities expanded dramatically and the differential methods have become themost commonly used methods for various complex and realistic geometries.6.4.9Smith–Spalding Integral MethodThe heat transfer in a constant-property laminar boundary layer with variable velocity U (x) but uniform surface temperature can be obtained via the Smith–Spaldingintegral method (1958), as described by Cebeci and Bradshaw (1984). A conductionthickness is defined asδc =k(T0 − T∞ )T0 − T∞=−q0(∂T /∂y)y=0This is expressed in nondimensional form asU (x) dδ2cδ2c dUδ2 dU=f, Pr = A − B cν dxν dxν dxBOOKCOMP, Inc.
— John Wiley & Sons / Page 466 / 2nd Proofs / Heat Transfer Handbook / Bejan(6.67)(6.68)HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW123456789101112131415161718192021222324252627282930313233343536373839404142434445467where A and B are Prandtl number–dependent constants. For similar laminar flows,the Nusselt number is given asNu =q xx(∂T /∂y)0=−= f (Pr,m,0)Re1/2xk(T0 − T∞ )T0 − T∞(6.69)and using the definition of the conduction thickness yieldsνxC2Uδ2c =(6.70)where C = f (Pr, m, 0). The parameters in eq. (6.68) can be expressed asU (x) dδ2c1−m=ν dxC2(6.71a)δ2cmdU= 2ν dxC(6.71b)Lines: 1339 to 1418Equation (6.69) is a first-order ordinary differential equation that can be integratedasνAδ2c =xUB−1dx+ δ2c0UBUiBUBc1 (U ∗ )c2q k== ρcp U (T0 − T∞ )ρcp U δx∗(U ∗ )c3 dx ∗1/21ReL1/2(6.72)0Herec1 = Pr−1 · A−1/2 ,c2 =B,2c3 = B − 1and provided in Table 6.1 andU∗ =U (x)U∞x∗ =xLReL =U∞ Lνwhere U∞ is a reference velocity, typically the uniform upstream velocity, and L is alength scale characteristic of the object.As an example of the use of the Smith–Spalding approach, consider the heattransfer in cross flow past a cylinder of radius r0 heated at a uniform temperature.The velocity distribution outside the boundary layer is given byBOOKCOMP, Inc.
— John Wiley & Sons / Page 467 / 2nd Proofs / Heat Transfer Handbook / Bejan———1.08134pt PgVar———Normal PagePgEnds: TEXwhere the subscript i denotes initial conditions. The normalized heat transfer coefficient in the form of a local Stanton number (St = Nu/Rex · Pr) isSt =[467], (29)[467], (29)468TABLE 6.1NumberParameters c1 , c2 , and c3 as Functions of the PrandtlPrc1c2c30.701.005.0010.00.4180.3320.1170.0730.4350.4750.5950.6851.871.952.192.37Source: Cebeci and Bradshaw (1984).U (x) = 2U∞ sin θ = 2U∞ sinxr0(6.73)where x is measured around the circumference, beginning with the front stagnationpoint.
For small values of θ, sin θ ≈ θ, and the free stream velocity approaches thatfor a stagnation point where similarity exists. The computed results for Nux /Re1/2xare shown in Fig. 6.11 for three values of Pr.[468], (30)Lines: 1418 to 1436———1.97408pt PgVar———Short PagePgEnds: TEXPr = 101.2Pr = 51[468], (30)0.8Nux公Rex123456789101112131415161718192021222324252627282930313233343536373839404142434445FORCED CONVECTION: EXTERNAL FLOWS0.6Pr = 0.70.40.20020406080180 x (deg) r0100120Figure 6.11 Local heat transfer results for crossflow past a heated circular cylinder at uniformtemperature for various Pr values, using the integral method. (From Cebeci and Bradshaw,1984.)BOOKCOMP, Inc.
— John Wiley & Sons / Page 468 / 2nd Proofs / Heat Transfer Handbook / BejanHEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW1234567891011121314151617181920212223242526272829303132333435363738394041424344456.4.10469Axisymmetric Nonsimilar FlowsThe Smith–Spalding method (1958) can be extended to axisymmetric flows by usingthe Mangler transformation. If the two-dimensional variables are denoted by the subscript 2 and the axisymmetric variables by the subscript 3 and neglecting transversecurvature, then r 2K r K00dx3θ2 =(6.74)θ3dx2 =LLwhere K is the flow index used in the Mangler transformation to relate the axisymmetric coordinates x and θ to two-dimensional coordinates and x3U B−1 dx3νA022r0 (δc )3 =(6.75)UBwhere r0 is the distance from the axis to the surface (see Fig.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
